Euclid: Elements

Nearly nothing is known of Euclid, except that he lived in Alexandria, Egypt, during the reign of Ptolemy I (323 BC – 283 BC). He was active at the great Library of Alexandria, and may have studied at Plato’s Academy in Athens, Greece.

The Elements consist of 13 books, some of which deal with domains of mathematics that today we call algebra and number theory, but the ancient Greeks treated these subjects always from a geometrical perspective. The Elements treat such important topics in planar geometry as: the Pythagorean theorem, equality of triangles, angles and their relation to triangles, angles and triangles inscribed in circles, tangents, circumscribed circles, polygons, Thales’ theorem, the golden ratio, and more.

Finally, in spacial geometry, topics include perpendicularity and parallelism in 3 dimensions, areas and volumes of parallelepipeds, cones, pyramids, cylinders, prisms, and the sphere, the regular (Platonic) solids inscribed in a sphere, and more.

Besides being a basic reference for geometrical and number-theoretical knowledge, another, most important contribution of the Elements is that it exemplified the axiomatic method and logical deduction (proving conclusions from premises), which became part of the subconscious of Western thought in later times. When today we say “Can you prove it?”, what we mean is, ideally, to start from some unquestionable assumptions (e.g., hard facts), and, making only logical deductions, prove the consequent proposition in the same way as propositions are proven in the Elements. In practice, of course, this can almost never be done in a mathematically perfect manner, but it is the idealization of this procedure that was inculcated in the minds of Western scholars and thinkers, in large part due to the Elements.

Euclid’s proofs are not error-free, because in some cases he makes use of propositions that appear “obvious”, but which he never stated (or proved) explicitly. This, however, is unavoidable, as anyone who has worked in automated theorem proving knows: if one wants to provide a complete proof of anything but the simplest statements, the true number of propositions on which the proof must rest explodes to proportions unmanageable by the human mind — hence only machines can keep track of some proofs in their entirety. Euclid used shortcuts, as every schoolteacher who teaches geometry inevitably must do.
 

Why is the text printed in this funny fashion, with letters appearing to “push” each other, until they all fall in place?

That behavior is observed only in the Java applet of this page (not in the stand-alone program), and occurs because each letter is an image (a GIF). Web browsers (as well as the Java engine that’s responsible for displaying those GIFs on your screen) show images in a random order when they have to show more than one of them on the same page. Thus, character images appear on a first-loaded-first-displayed basis, until they are all in place. However, once an image is loaded it doesn’t need to be reloaded when re-printed, unless you jump out of this page; that’s why this funny behavior diminishes as you scroll down the text. However, this problem does not exist in the stand-alone program, because that one runs locally on your computer.

Why is this weird font used for classical texts? Why does the circumflex look strange? Why don’t the letters look more like the usual ancient Greek letters of classical literature or the Bible?

The font employed by Classics Reader is the standard Times Roman font used in Greek literature in Greece, to print both modern and classical texts. The circumflex is correct according to that tradition. The reader who wonders about this issue is probably more familiar with fonts used in western Europe and the USA to print classical and biblical Greek texts. There is no reason why the latter tradition should be preferred over the former. Note that west European and American fonts for ancient Greek texts cannot be “closer to the original”, because ancient texts were hand-written by scribes, so their appearance depended on the scribe’s handwriting; it was only relatively recently that typography standardized what is now recognized as “ancient Greek font” in the West.

Sometimes, when using the scrollbars of the browser (not the Classics Reader’s own ones), some line or lines appear misprinted in the Reader’s text.

This is a problem created by the browser, and it appears occasionally when scrolling up or down, while a Java applet attempts to paint in its own rectangle. To refresh the text, click on the Chapter selection (the middle of the three choice-boxes) and select the same chapter that you are currently reading. Once again, this problem does not exist in the stand-alone program.